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Inverse

In Set Theory, given functions \(f:X\to Y\) and \(g:Y\to X\), one says that \(g\) is the `inverse' of \(f\) (written \(g=f^{-1}\) if \(f\circ g=\mathrm{id}_Y\) and \(g\circ f=\mathrm{id}_X\). In a general Category \(\mathcal{C}\), an arrow \(f:X\to Y\) in \(\mathcal{C}\) is called invertible if there exists an arrow \(g:Y\to X\) in \(\mathcal{C}\) such that \(f\circ g=\mathrm{id}_Y\) and \(g\circ f=\mathrm{id}_X\). Such an arrow is called an `isomorphism' in \(\mathcal{C}\).